Inclusion of the backaction term in the total optical force exerted upon Rayleigh particles in nonresonant structures

In this paper, we investigate the impact of the electromagnetic scattering caused by other objects in nonfree space on the time-averaged force exerted upon a Rayleigh particle, which is conventionally referred to as the backaction effect. We show that backaction modifies the gradient force, radiation pressure, and spin curl force exerted upon Rayleigh particles, and gives rise to an additional force term which stems from the gradient of the backaction field in nonfree space. As a numerical example, we look into the trapping of a dielectric nanoparticle at the center of curvature of a spherical mirror, and study how it is affected by the backaction effect. We show that backaction can enhance the force exerted upon the particle, reshape the trapping potential, and shift the equilibrium position of the particle.

Figure 1

Schematic of the system. A dielectric nanosphere is trapped by an optical tweezer at the center of a spherical mirror.

Figure 2

Optical force exerted upon a 110-nm nanosphere with refractive index 2.5 at the $z=0$ plane. (a),(b) For $R_0=(2n-1)\frac{{\lambda }_0}{4}$. (c),(d) For $R_0=(2n+1)\frac{{\lambda }_0}{4}$. (e),(f) In the absence of the spherical mirror. These results are obtained for $I_0=2\left[\frac{\text{mW}}{\mu {\text{m}}^2}\right]$ and $w_0=3{\lambda }_0$.

Figure 3

Trapping potential. (a) Along the $x$ axis for $R_0=(2n-1)\frac{{\lambda }_0}{4}$. (b) Along the $x$ axis for $R_0=(2n+1)\frac{{\lambda }_0}{4}$. (c) Along the $y$ axis for $R_0=(2n-1)\frac{{\lambda }_0}{4}$. (d) Along the $y$ axis for $R_0=(2n+1)\frac{{\lambda }_0}{4}$. The red-dashed lines show the backaction free results obtained in the absence of the spherical mirror.

Figure 4

(a) Imaginary part of the polarizability normalized to the imaginary part of the free-space polarizability along the $z$ axis. (b) Force exerted upon the nanosphere along the $z$ axis. The red-dashed line is the force in the absence of the spherical mirror. The blue-dotted line is the scattering force in the presence of the nanosphere neglecting the contribution of the gradient of the scattering Green's function. These results are obtained for $R_0=(2n-1)\frac{{\lambda }_0}{4}$.

Figure 5

(a) Schematic of the CNH structure. (b) Transmission spectrum of CNH structure in the absence of the nanosphere. (c), (d) Intensity profiles of the electromagnetic fields in the absence of the nanosphere.

Figure 6

(a) Optical force exerted upon a 50-nm-radius polystyrene nanoparticle along the $x$ direction at the entry of the aperture. The force is calculated with three different methods: Maxwell stress tensor (red-dotted), dipole approximation including the backaction effect (solid black), and conventional dipole approximation without considering the backaction effect (blue-dashed). (b) Contribution of different force terms. (c) Real part of the polarizability of the nanosphere normalized to the polarizability of the nanosphere in the free space.

Figure 7

Optical force exerted upon a 50-nm-radius polystyrene nanoparticle along the $z$ direction at (a) $x=100$ nm and (b) $x=90$ nm from the center of the hole. (Red-dotted) Maxwell stress tensor, (solid black) dipole approximation including the backaction effect, (blue-dashed) conventional dipole approximation without considering the backaction effect.